Fano hypersurfaces and Calabi-Yau supermanifolds

TL;DR

This paper explores the geometric structures of supervarieties linked to N=2 Landau-Ginzburg models and Calabi-Yau supermanifolds, testing conjectures about their singularities and establishing links to complete intersection Calabi-Yau manifolds.

Contribution

It provides a geometric interpretation of supervarieties associated with Gepner models and tests a conjecture on their superdimension and singular locus.

Findings

Supervarieties have vanishing super-first Chern class.

Fano hypersurfaces can be linked to Landau-Ginzburg orbifolds.

Nef partition of the Newton polytope enables geometric interpretation.

Abstract

In this paper, we study the geometrical interpretations associated with Sethi's proposed general correspondence between N = 2 Landau-Ginzburg orbifolds with integral \hat{c} and N = 2 nonlinear sigma models. We focus on the supervarieties associated with \hat{c} = 3 Gepner models. In the process, we test a conjecture regarding the superdimension of the singular locus of these supervarieties. The supervarieties are defined by a hypersurface \widetilde{W} = 0 in a weighted superprojective space and have vanishing super-first Chern class. Here, \widetilde{W} is the modified superpotential obtained by adding as necessary to the Gepner superpotential a boson mass term and/or fermion bilinears so that the superdimension of the supervariety is equal to \hat{c}. When Sethi's proposal calls for adding fermion bilinears, setting the bosonic part of \widetilde{W} (denoted by \widetilde{W}_{bos}) equal to zero defines a Fano hypersurface embedded in a weighted projective space. In this case, if the Newton polytope of \widetilde{W}_{bos} admits a nef partition, then the Landau-Ginzburg orbifold can be given a geometrical interpretation as a nonlinear sigma model on a complete intersection Calabi-Yau manifold. The complete intersection Calabi-Yau manifold should be equivalent to the Calabi-Yau supermanifold prescribed by Sethi's proposal.